Suppose v x are vectors in V and U W are subspaces of V such

Suppose v, x are vectors in V and U, W are subspaces of V such that v + U = x + W. Prove that U = W.

Solution

Let a basis for V be { v₁ , v₂ , ...,v_n }.Since v and x are vectors in V, there exist scalara a₁, a₂,,,,, a_n and b₁ , b₂ , ..., b_n not all 0, such that v = a₁v₁ + a₂v₂ + ...+a_nv_n and x = b₁v₁ + b₂v₂ +... + b_nv_n. Also, let { u₁ , u₂, ...u_p } and { w₁ , w₂ , ..., w_q } be bases for U and W. Then, since, v + U = x + W, therefore, a₁v₁ + a₂v₂ + ...+a_nvn + U = b₁v₁ + b₂v₂ +... + b_nv_n + W or, (a₁- b₁) v₁ + (a₂ -b₂)v₂ +...+(a_n -b_n)v_n + U = W. Therefore, U is a subsapace of W, as a_i -b_i 0 for all i\'s otherwise v and x will be same. Similarly, ( b₁ - a₁)v₁ + (b₂ - a₂ )v₂ +...+ (b_n -a_n)v_n +W = U so that W is a subspace of U. Hence U = W.